b) develop mathematical thinking and problem solving ability.

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1 Submission for Pre-Calculus MATH Course s instructional goals and objectives: The purpose of this course is to a) develop conceptual understanding and fluency with algebraic and transcendental functions, including exponential, logarithmic and trigonometric functions, techniques, and manipulations necessary for success in Calculus. b) develop mathematical thinking and problem solving ability. 2. Assessment and/or evaluation of student learning in the course: Students have 6 exams, a comprehensive final exam, and several quizzes during the semester. Students will complete homework on an online course management system and this homework is graded. Class participation is also used as a formative assessment, but most often is not graded. Learning Outcomes for Precalculus: 1 * Learning Outcome Description 1.0 Functions 1.1 Represent functions verbally, numerically, graphically and algebraically, including linear, quadratic, polynomial, rational, root/radical/power, exponential, logarithmic and piecewise-defined functions. 1.2 Determine whether an algebraic relation or given graph represents a function. Describe how the Courses or Programs address each outcome Students will recognize from the graph of a relation whether it is a function by applying the vertical line test. They will determine the domain of a function from its graph, from its symbolic representation, and in context of applications. Describe how much time the course spends on this topic 14 Days

2 1.3 Perform transformations of functions translations, reflections and stretching and shrinking. 1.4 Perform operations with functions addition, subtraction, multiplication, division and composition. 1.5 Analyze the algebraic structure and graph of a function, including those listed in (1.1), to determine intercepts, domain, range, intervals on which the function is increasing, decreasing or constant, the vertex of a quadratic function, asymptotes, whether the function is one-to-one, whether the graph has symmetry (even/odd), etc., and given the graph of a function to determine possible algebraic definitions. 1.6 Find inverses of functions listed in (1.1) and understand the relationship of the graph of a function to that of its inverse. A graphing utility will be used to explore translation, reflection, stretching and shrinking by corresponding adjustment of coefficients, though students will eventually be asked to graph these by hand without access to a calculator. After manually determining intercepts, domain, symmetry, and asymptotes (if applicable) students will graph polynomial and rational functions by hand. Theory of equations, e.g. synthetic division, will be applied to polynomials to locate roots between consecutive integers. Students will explore adding, subtracting, multiplying, dividing, and forming the composite of functions by comparing the results obtained from the new functions with those obtained from the component functions. Comparisons will be made numerically, graphically (using a graphing utility) and symbolically. Students will solve real-life situations by choosing an appropriate mathematical model and solving. They will be asked to report the effects on the outcome of changing the variables in the model to reflect possible changes in the real-life situation. 1.7 Use the Remainder and Factor Theorems for polynomial functions. 1.8 Use functions, including those listed in (1.1), to model a variety of real-world problem solving applications. 2 * 2. Equations/Systems 20 Days 2.1 Understand the difference between an algebraic equation of one, two or more variables and a function, and the relationship Students will write possible functions given a given graph or x-intercepts and then check their work with a graphing utility.

3 among the solutions of an equation in one variable, the zeros of the corresponding function, and the coordinates of the x-intercepts of the graph of that function. 2.2 Determine algebraically and graphically whether the graph of an equation exhibits symmetry. 2.3 Solve a variety of equations, including polynomial, rational, exponential, and logarithmic, including equations arising in application problems. 2.4 Solve a system of linear equations graphically and algebraically by substitution and elimination, and solve application problems that involve systems of linear equations. 2.5 Identify and express the conics in standard rectangular form, graph the conics, and solve applied problems involving conics 2.6 Solve polynomial and rational inequalities graphically and algebraically. Students will use the laws of exponents and logarithms to solve exponential and logarithmic equations. They will choose appropriate variations of these two functions to model and solve real-life problems. Students will discover through composition of functions that exp(log (x)) = x and log(exp(x)) = x and thus exp(x) and log(x) are inverse functions. They will use a graphing utility to observe that the graphs are symmetric with respect to the line y = x. Students will hunt for the zeros of polynomials by following the clues obtained from The Remainder Theorem, The Factor Theorem, and the Rational Roots Theorem. They will apply synthetic division to locate roots between consecutive integers. The rigor of these techniques will help to develop their analytical skills and lead to an appreciation of the graphing calculator techniques by comparison. The Factor Theorem will lead to an understanding of the relation between roots and factors. The students will learn to view the circle, parabola, hyperbola, and ellipse as intersections of a plane with a cone. They will obtain equations of the conic sections from definitions in terms of loci of points and they will use the characteristics of each to sketch the graphs manually. They will also discover how the properties of the conics have been used in architecture and engineering and they will solve problems regarding these reallife situations. Students will learn to apply fundamental principles of parity and sign analysis diagrams to the solution of polynomial and rational inequalities. They will discover how to solve the same inequalities visually by analyzing the graphs on a graphing utility.

4 3 3.1 Represent sequences verbally, numerically, graphically, and algebraically, including botht the general term and recursively. 3.2 Write series in summation notation, and represent sequences of partial sums verbally, numerically and graphically. 3.3 Identify and express the general term of arithmetic Express general terms of various sequences (e.g., arithmetic and geometric), write series in summation notation, find the sum of arithmetic and geometric series. Students will explore the language and symbols of summation and learn to obtain large sums through the use of formulas later to be applied to the Calculus. Some methods of proof such as mathematical induction may be included to improve mathematical reasoning skills. 4 days Trigonometric Functions 1.1 Represent trigonometric and inverse trigonometric functions verbally, numerically, graphically and algebraically; define the six trigonometric functions in terms of right triangles and the unit circle. 1.2 Perform transformations of trigonometric and inverse trigonometric functions translations, reflections and stretching and shrinking (amplitude, period and phase shift). 1.3 Analyze the algebraic structure and graph of Students will first learn the ratio and the unit circle definitions of all trigonometric functions. Then they will understand how to recognize a given trigonometric function from the graph or formula. Students will sketch by hand graphs involving translations, reflections, stretching and shrinking by corresponding adjustment of the amplitude, period, phase shift and vertical translation. Pythagorean theorem, the distance and midpoint formulae, together with the Algebra for Calculus knowledge of transformations, symmetry, and inverse functions will be applied to recognize the stretching and shrinking of trigonometric functions. Students will explore adding, subtracting, multiplying, dividing, and forming the composite of trigonometric functions by using their Algebra for Calculus knowledge together with comparing the 7 days

5 trigonometric and inverse trigonometric functions to determine intercepts, domain, range, intervals on which the function is increasing, decreasing or constant, asymptotes, whether the function is one-to-one, whether the graph has symmetry (even/odd), etc., and given the graph of a function to determine possible algebraic definitions. 1.4 Use trigonometric and inverse trigonometric functions to model a variety of real-world problemsolving applications. results obtained from the new functions with those obtained from the component functions. Students will solve real-life situations and problems by choosing an appropriate mathematical repetitive motion model and finding forces acting on parts Trigonometric Equations. 2.1 Solve a variety of trigonometric and inverse trigonometric equations, including those requiring the use of the fundamental trigonometric identities in degrees and radians for both special and non-special angles. Solve application problems that involve such equations. Students will use the basic sine, cosine, and tangent equations to solve conditional trigonometric equations. They will choose appropriate variations of the six trigonometric functions to model and solve real-life problems. Students will discover the area of triangles through the laws of sine and cosine. They will use a graphing utility to observe that the graphs of sine and tangent are symmetric with respect to the line y = x, and the graphs of cosine and cotangent are symmetric with respect to the y-axis. Students will discover how to solve the trigonometric equations visually by analyzing the graphs on a graphing utility. 5 days 3 Angles/triangles 5 days 3.1. Express angles in both degree and radian measure Solve right and oblique triangles in degrees and radians for both special and non-special angles, and solve application problems that involve right and oblique triangles. Students will find trigonometry as a powerful tool for finding areas of plots of land, lengths of sides, and measures of angles, without physically measuring them. Students will learn the notions of Radian Measure, Arc Length, and Area through the solutions of problems about finding the area of a sector of a circle and the length of the circle. They will apply their knowledge to real life problems about linear and angular velocity.

6 They will also learn the problems that involve the right and oblique angles by solving problems about the angle of elevation and angle of depression as well as finding the height of an object from a distance. 4 Trigonometric identities 4.1. Verify trigonometric identities by algebraically manipulating trigonometric expressions using fundamental trigonometric identities, including the Pythagorean, sum and difference of angles, double-angle and half-angle identities. 5 Vectors Represent vectors graphically in both rectangular and polar coordinates and understand the conceptual and notational difference between a vector and a point in the plane. 5.2.Perform basic vector operations both graphically and algebraically addition, subtraction and scalar multiplication Solve application problems using vectors. Students will learn the sum and difference identities for sine, cosine, and tangent, double-angle and half-angle identities, as well as the product and the sum identities by solving problems containing these identities. They will also apply their knowledge through the problems of modeling of the motion of a spring. Students will learn how to find the horizontal and vertical components of vectors, their magnitude, direction, as well as the angle between vectors by solving inclined-plane problems, the problems of finding the weight of an object, and navigation problems. To solve these problems they will have to apply all of the previous knowledge they obtained during the course trigonometry. 7 days 6 days